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Model Theory and Differential Equations

模型理论和微分方程

基本信息

批准号：
1700095
负责人：
James Freitag
金额：
$ 14.71万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-06-01 至 2020-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700095&HistoricalAwards=false
关键词：
Model Theory Differential Equations

项目摘要

This project centers on algebraic differential equations and model theory. Differential equations are at the heart of modern mathematics and its applications in other sciences. Model theory is a part of mathematical logic in which objects known as definable sets are studied. The notion of a definable set is flexible, changing as one works in different mathematical settings. In the area of differential algebra, definable sets are closely linked with differential equations, providing a connection between the two subjects. This project plans to exploit that link to advance the knowledge of both fields. The project also seeks applications in other areas of mathematics, primarily number theory.This research project addresses problems in the model theory of fields with operators. First, the investigator will generalize his work on the differential algebra of the j-function to understand the differential equations satisfied by analytic covering maps associated with various Shimura varieties. This line of work is expected to have number theoretic consequences related to special-point conjectures as it did in the case of modular curves. The project will also investigate several finiteness results in differential algebra; this line of work is expected to have consequences on effective bounds in number theory and in computational differential algebra. The project aims to develop various aspects of the model theory of supersimple and superstable groups as a generalization of recent work in differential algebraic and difference algebraic groups. The project will investigate the classification of disintegrated strongly minimal sets in differentially closed fields. This work seeks to answer a fundamental question about differential equations: what are the possible structures given by the differential algebraic relations between solutions to a fixed differential equation? Finally, this project aims to adapt and generalize results from algebraic foliations and vector fields for application in differential algebraic geometry.

该项目以代数微分方程和模型理论为中心。微分方程是现代数学及其在其他科学中应用的核心。模型论是数理逻辑的一部分，其中研究称为可定义集合的对象。可定义集合的概念是灵活的，随着人们在不同的数学环境中工作而变化。在微分代数领域，可定义集与微分方程密切相关，在两个学科之间提供了联系。该项目计划利用这一联系来推进这两个领域的知识。该项目还寻求在其他数学领域的应用，主要是数论。该研究项目解决了算子域模型论中的问题。首先，研究者将概括他在 j 函数微分代数方面的工作，以理解与各种 Shimura 品种相关的解析覆盖图所满足的微分方程。预计这一工作将产生与特殊点猜想相关的数论结果，就像在模曲线的情况下一样。该项目还将研究微分代数中的几个有限性结果；这一工作预计将对数论和计算微分代数的有效界限产生影响。该项目旨在发展超简单群和超稳定群模型理论的各个方面，作为微分代数和差分代数群最近工作的概括。该项目将研究微分闭域中可分解的强极小集的分类。这项工作旨在回答有关微分方程的一个基本问题：固定微分方程解之间的微分代数关系给出的可能结构是什么？最后，该项目旨在调整和推广代数叶状结构和向量场的结果，以应用于微分代数几何。